Area of intrinsic graphs and coarea formula in Carnot groups
Antoine Julia, Sebastiano Nicolussi Golo, Davide Vittone
Abstract
Abstract We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math> regularity ( $$C^1_H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> ). Our first main result is an area formula for $$C^1_H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $$C^1_H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> submanifolds into level sets of a $$C^1_H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> function.